Bathroom fan fan hums when connected to the timer.

No, both cannot be connected to same vent.
Or vent exhause from 1 fan will vent into other bathroom.
All house vents require separate duct.

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The start capacitor for the fan motor is defective.

This could be the timer or the motor. The timer could be going out or not starting up in the right position. Has anyone worked or removed the timer knob, it could have been put on just a little off causing the switch to be off just a bit. Again the motor could be going out,such as the bearings inside. I would see what type of warranty is on this fan. I know these exhaust fans can get a little pricey, so I would see what warranty options I have.

Leviton countdown timers have additional wire that powers internal electronics.
Leviton timers have red-black-white-green wires (or blue-black-white-green wires)

1) Regular single pole switch has 2 insulated wires. One of these wires is Hot from breaker, and the other goes to Load (fan)
2) Timer green wire connects to bare ground as you know.
3) Hot from breaker connects to timer black wire.
4) Load wire connects to timer red (or blue) wire.

5) Located in back of box are white wires that are twisted together and covered with wire nut.
These are Neutral wires. Connect timer white to these wires. This will power timer electronics.
6) If box does not have Neutral wires described above, then connect timer white to bare ground along with timer green wire.

7) Turn power ON and try timer countdown.
If timer does not work, then reverse timer red and timer black wire.

8) If box does not have neutral wires: Image on link below shows similar Intermatic timer:
Image applies ONLY when Neutral is not available.
http://waterheatertimer.org/images/EI210-wiring-full.jpg

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You need "15 minute countdown timer" for a bath fan.

1) Following page shows selection of timer types and brands:
http://waterheatertimer.org/Countdown.html
There are push button electronic types and rotary dial types.

2) Easiest to wire is mechanical spring type countdown timer, FD series.
No electronics that can be affected by 'electric noise' on fan circuit.
Intermatic spring-wound last for years without problem.
http://www.electricsupplyonline.com/prod/intermatic-springwound_timers.php

http://www.electricsupplyonline.com/prod/intermatic-springwound_timers/fd15mc_w005485.php

3) Another reliable timer is GE 15088 sold at Home Depot.
Timer has preset for 20 minutes, not 15
http://www.homedepot.com/webapp/wcs/stores/

4) Amazon sells reliable 15 minute Intermatic electronic countdown:
http://www.1000bulbs.com/product/3293/ELEC-EI200LA.html?utm_source=SmartFeedAmazonProduct

5) Timex sells 15 minute spring wound countdown timer. Lower price is probably with reason since Timex are generally sold in drug store/dollar store rather than hardware/electric outlets.
http://www.amazon.com/Timex-Minute-Springwound-Countdown-Timer/dp/B00013K69Q

6) Spring wound timers connect easily to same wires that come off old fan switch.
Old switch has 2 insulated wires, connect them to either terminal on spring wound timer.

7) Electronic timers are wired differently:
Instructions for wiring electronic timers appears on following link:
http://waterheatertimer.org/GE-timers-and-manuals.html#15088

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Also take advantage of fixya phone service.
For a price, expert speaks with you over phone while you work on timer or any do-it-yourself project.
Fixya is always less expensive than a service call.

Sure George.

These in-wall timers have extra wire to operate the electronics located inside timer.
You said your old switch has 3 wires.
The new timer can only replace on-off switch that has 2 brass screws and a green screw.
If old switch has 2 brass screws and a dark screw, that is a 3-way switch.
Typically a 3-way switch is found at both ends of hallway.
The Woods 95008 countdown timer is not made to replace 3-way:
http://waterheatertimer.org/Woods-timers-and-manuals.html#59000

Here's what to do:
1) Switch has 2 wires (plus bare ground wire). Take switch wires and connect them to timer-black-wire and timer-red-wire. Choose either one for now.
2) In back of electric box are white wires twisted together and covered with wire nut. Connect timer-white to these wires. This wire will power timer electronics.
3) If electric box does not have white wires described above, then connect timer-white to bare ground, along with timer-green wire.
4) Turn on electricity, and push override button located below the OFF button. If lights come on, timer is wired correctly. If lights do not come on, then reverse timer-red and timer-black wires.

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First, be certain that the power is off. The black goes to black, the white goes to white and the red goes to the fan, it is the switched wire....

Switch has 2 black wires. One black is hot wire. One black is wire going to fan. Turn off power. Remove wires from switch and separate. Turn power back on. Using ordinary tester, test each black wire to bare copper ground wire. Only one of the black wires will test hot. The other black wire goes to fan.
Timer black wire connects to black hot
Timer blue wire connects to black wire going to fan
Timer white wire connects to white neutral (in absence of white neutral, connect Timer white wire to bare ground)
Timer green connects to bare ground wire

One needs to install a bathroom fan that has a Sone rating of 1 or below. They aren't cheap :-( Sorry, there really isn't any other solution.

bathroom fans require a mains permenant and a switched wire from pull switch for the timer to work